Multi-Sectoral Growth and Technological Change

Multi-Sectoral Growth and Technological ChangeAuthor(s): B. L. ScarfeSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 4, No. 3(Aug., 1971), pp. 299-313Published by: Wiley on behalf of the Canadian Economics AssociationStable URL: http://www.jstor.org/stable/133776 .

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MULTI-SECTORAL GROWTH AND TECHNOLOGICAL CHANGE*

B. L. SCARFE University of Manitoba

Croissance multisectorielle et changement technologique. L'auteur presente un modele de croissance a fonction de Cobb-Douglas et multisectoriel. I1 y tient compte du changement technologique, qui se produit a des rythmes differents suivant les secteurs. I1 passe en revue certaines des caracteristiques importantes de la croissance multisectorielle en 1'absence de changement technologique, particulibrement la question de la stabilite.

Le niveau de la technologie utilisee dans chaque secteur evolue suivant une fonction de forme logistique, comme dans plusieurs processus de diffusion. L'auteur montre que meme quand le changement technologique prend ainsi une forme definitive qui conduit a une nouvelle situation de croissance reguliere (. steady-state ,), les valeurs des stocks de capital, des prix relatifs et des salaires reels qui y correspondent sont differentes de celles de la situation de depart. Les ameliorations technologiques conduisent a des changements des prix relatifs et par suite A des effets de substitution qui prennent la forme d'un processus de reallocation du capital. C'est ainsi qu'une economie s'adapte a une nouvelle technologie. Ce processus de transition se produit en meme temps que l'accumulation du capital. C'est par ce dernier mecanisme que s'effectue le transfert de l'augmentation des taux de rendement associee aux ameliorations technologiques en faveur des salaries, sous forme d'un accroisse- ment des taux de salaires reels.

I / Introduction

Although modern growth models stress that increases in labour productivity depend upon both capital accumulation and technological progress, recent multi-sectoral formulations of growth theory concentrate mainly on equilibrium paths of capital accumulation to the neglect of technological progress. This paper outlines a method by which the impact of technological change on a multi-sectoral economy may be analyzed without excessive complication, and, as a by-product, reviews some basic features of multi-sectoral growth in the absence of technological change. The basic model considered has n + 1 productive sectors which use combinations of n capital goods (commodities 1 to n) and labour to produce the n capital goods themselves and a consumption good (commodity 0), the consumption good always being taken to be the numeraire. The underlying technology is taken to be of the Cobb-Douglas form, withl constant returns to scale. Depreciation (which is independent of the sector in which a capital good is used, but not of the type of capital good) occurs by exponential decay so that new and old capital goods of a given type are indistinguishable and production, although using fixed capital, is non-joint. The concern throughout is with equilibrium paths on which (a) all capital goods and labour are fully employed at all points of time, and (b) prices are

*This paper is based on a part of the author's unpublished D.Phil. dissertation; "Capital Accumulation and Comparative Advantage: A Critical Appraisal" (Oxford University, 1970). The author is indebted to W. M. Gorman, C. W. Ekstrand, and the referees for valuable comments and advice.

Canadian Journal of Economics/Revue Canadienne d'Economique, IV, no. 3 August/ao'ut 1971



300 B. L. SCARFE

equal to costs at all points of time, where costs are such that the reward to each input is the same wherever that input is used.

For such a model, two sets of initial conditions are required in order to set a process of capital accumulation in motion. These are (a) the initial stock positions or capital-labour ratios, and (b) the initial expectations of the proportional rates of change of the capital-goods prices. The model is closed by postulating that the labour supply grows at the steady-state rate, y, and that all net savings come from net profits (with depreciation allowances being wholly saved) so that, if a steady-state exists, the common steady-state rate of profit on all capital goods, r, is uniquely determined by , = s,r, where O < sr < 1 is the savings propensity out of profits.

In the following two sections of this paper, the formal structure of the model is outlined and a review of its basic properties in the absence of technological progress is undertaken. The subsequent sections introduce disembodied technological change, which is allowed to occur at different rates in different sectors. The resulting changes in relative prices generate a process of investment and disinvestment, or capital transmutation, which alters the relative specification of the economy's capital stock vector.

II | The formal structure of the model

The basic n + 1 Cobb-Douglas production functions may be written in the form

n

(1) ln Yj(t) = boj ln tj(t) + boj ln Lj(t) + j bij ln Kij(t), j = 0,1 ... n n n

with L(t) = j Lj(t), and Ki(t) = , Kij(t), i = 1 ... j=0 j=0

where Yj(t) is the output of commodityj at time t, Kij (t) is the input of class i capital goods in the production of commodity j at time t, with Ki (t) being the fully-employed stock of class i capital goods at time t, Lj(t) is the labour input in the production of commodity j at time t, with L (t) being the fully-employed stock of labour at time t. The boj, j = 0, 1 ... n, and the b1j, i = 1 ...n, j = 0, 1 ... n, represent labour and capital good input-elasticities which obey the conditions boj + Es-l bij = 1 for all j = 0, 1 ... n, indicating constant returns to scale. t (t) is a shift parameter representing the possibility of Harrod-neutral technological progress in sector j at time t, and ln is the logarithmic operator. In competitive equilibrium, the marginal productivity conditions may be written as

(2) w(t)Lj(t) = bojpj(t)YjW(t),j = 0, 1 ... n, and

pi(t){ri(t) + d1(t)}Kij(t) = bijpj(t)Yj(t), i = 1 ... n,j = 0,1 ... n,

where pi(t) is the price of the ith commodity at time t (with the consumption good as numeraire so that po(t) = 1 for all t), w(t) is the real wage rate at time t, ri(t) is the net rate of return on capital good i at time t, and di(t) is the proportion of the existing stock of capital good i which "evaporates" or



Multi-Sectoral Growth 301

depreciates in production (which will herein be assumed constant for all I). Using these conditions to eliminate Lj(t) and Kij(t) in (a) the factor balance equations of expression 1 and (b) the production functions themselves, one obtains the following set of primal and dual equations (with time subscripts omitted for convenience):

n n

(3) wL = bojpjY (r1 + dj)pjK1 E bijpjYj, i = 1 ... n, and .?=0 .1=?

n

0 = a0 + boo ln(w/to) + > ln{pi(ri + dj)}bjo,

ln pj aj + boj ln(w/ij) + ln{pi(r, + di)}bij, j = I

Ein~~~~~= where a1 = - o bi1 In b1j, j = O, 1 ... n. Using the substitutions (a) Y = (gj + dj)Kj, j = 1 ... n, where gj is defined to be the net growth rate

of the stock of capital good j, and (b) c- Yo/L, ki K,/L, where c is con- sunmption per head and ki is the ith capital-labour ratio, one may re-write the factor-balance equations of expression 3 as

2%

(4) w = booc + E boj(gj + dj)pjkj, and ,j=1

(rj + di)piki = bioc + Z bi(gj + dj)pjkj, i = 1 ... n. j==1

Eliminating c across the equations 4 and w across the price equations of expression 3, and expressing the result compactly in matrix notation, one has the fundamental primal and dual equations

(5) { (R + D) + B(G + D)}Pk = b,boo-1w, and (I + B') ln(p) + B' ln (R + D)i = a -In()bwy

where k is an n X 1 column vector of capital-labour ratios, p is an n X 1 column vector of capital goods prices with P being the corresponding n X n diagonal matrix, G, R, and D are n X n diagonal matrices containing the n net growth rates, the n net rates of return, and the n depreciation coefficients, respectively, I and i are the n X n unit matrix and the n X 1 unit vector, respectively, t is an n X n diagonal matrix with elements J j1 = ... ny a is an n X 1 column vector with elements aj - aoboo-lbo1 j = 1 ... n, B is an n X n matrix with elements boboo-lboj - bij, all i,j = 1 ... n, B' being its transpose, b, is an n X 1 column vector of capital input-elasticities into the consumption-goods sector, and bw is an n X 1 column vector of labour input- elasticities into the capital-goods sectors.

To complete the system one requires a set of n accumulation equations and a set of n own-rate of interest equations of the form:

(6) k = (dk/dt) = (G - JI)k, and -p = - (dp/dt) = (R - rI)p.

The first n of these equations - the accumulation equations - indicate that the proportional rate of growth of the ith capital-labour ratio, kt(t)/kj(t), is equal to the rate of growth of the stock of capital goods of type i, gi (t), less the rate



302 B. L. SCARFE

of growth of the labour supply, A. The second n of these equations - the own- rate of interest equations - indicate that the proportional rate of change of the price of the ith capital good, Pi (t)/pi (t), plus the net rate of return on that capital good, ri(t), must be equal to a common interest rate, r(t), for all capital goods. Notice that these latter equations embody the "perfect foresight" assumption that expectations are always realized since the equations are assumed to hold ex post as well as ex ante. (This "perfect foresight" assumption will later be replaced by a "zero foresight" assumption.) Finally, the rate of interest is determined from the savings-investment equilibrium condition which boils down to the classical g (t) = srr (t), where g (t) = g1 (t)p1 (t)K1 (t)/

Ai-jpi(t)Ki(t) is the over-all growth rate of the capital stock.

III / Some basic properties of the model

The formal structure of the model has now been outlined. Some of its basic properties in the case in which there is no technological progress - which may be represented formally by the assumption that each and every tj(t), j = 0, 1 ... n, is equal to unity for all time t so that t is the n X n identity matrix - can be described as follows. First, it can be shown' that a momentary equilibrium satisfying the Cobb-Douglas system (equations 5 and 6) with positive prices and quantities will exist for any given initial capital-labour ratios and any given initial price expectations. Such an equilibrium may be represented by a point on an n + 1 dimensional factor-price frontier which relates the real wage rate, w (t), to the net rates of return on each of the n types of capital goods, r,(t), i = 1 ... n. For convenience, this frontier may be expressed in the following way:

(7) w(t) = to(t)Q{R(t), (t)}.

Its exact mathematical expression is, however, given by

(8) ln(w/to) = -aoboO-1 + b'1{I - B'WC}-1{ln(t)bw - ln(R + D)i -a,

where BWC is an n X n matrix with elements b11, i = 1 ... n, j = 1 ... n, B'w, being its transpose, giving B = boo-lb-bw - B, Notice that on the usual assumptions Bw, is an indecomposable non-negative matrix with all of its column sum less than unity (labour being required as an input in every productive sector). Thus I - BW is a Leontief matrix so that (I - B'w,)-l exists and is positive. It follows that the factor-price frontier defines w (t) as a strictly positive differentiable function of R (t) which is strictly monotonically decreasing in each of its arguments, ri(t), i = 1 ... n. At any time t, this frontier specifies the maximal real wage rate that is achievable at each feasible configuration of net rates of return. Figure 1 graphs the factor-price frontier for the case in which there are only two capital goods.

Secondly, momentary equilibrium need not be unique. Since non-uniqueness of momentary equilibrium at any point of time implies that the system may 'See F. H. Hahn, "Equilibrium Dynamics with Heterogeneous Capital Goods," Quarterly Journal of Economics, 80 (Nov. 1966), 638-40.




locus rl=iO I asymptotic to I

locus r2= \d2 locus r2 0

2 /\asymptotic to locus r1 = d

L/ # \locus r1 r2

-dl r--

locus W== Wmin >0

r2/

FIGURE 1 The factor-price frontier

follow alternative trajectories from that point of time onwards so that the whole future path of the system is not unique, it is useful to specify a set of conditions which are sufficient for uniqueness. It is intuitively obvious that momentary equilibrium will be unique if the relationship between p(t) and R (t) is globally one-for-one. This will be so if p (t) is a strictly monotonic function of w(t), since it is known from the factor-price frontier that w(t) is strictly monotonically decreasing in each ri(t), i = 1 ... n. Differentiating the price equations of expression 5 with respect to w(t) one has

(9) (I + B')P-1(dp/dw) + B'(R + D)-1(dR/dw)i = 0.

Now since dR/dw is a diagonal matrix with strictly negative diagonal elements, it follows that dp/dw will be one-signed if the matrix (I + B')-'B' has one- signed elements. In particular, dp/dw will be strictly positive if (I + B')-'B' is a non-negative matrix with every row being semi-positive. Two alternative sufficiency conditions for this to be so are (a) that B be a non-negative matrix with every column being semi-positive, so that the production of the consumption-good always uses at least as much of every capital-good (and more of some) relative to labour as the production of every capital-good uses, and



304 B. L. SCA1FE

(b) that I + B-' have a dominant positive diagonal and non-positive off- diagonal elements, thus being a Leontief matrix. These two alternative sufficiency conditions are, of course, both pretty stringent, though there is no necessary connection between them. Momentary equilibrium will, therefore, be unique if the relative price, pi(t), i = 1 ... n, of each and every capital-good always increases as the wage rate, w(t), increases, that is, if the consumption- good is unambiguously the most capital-intensive commodity, a generalized capital-intensity condition.

Provided that a unique momentary equilibrium exists at every point of time along a warranted path of capital accumulation it is useful to consider the asymptotic properties of such a path. In particular, since it can easily be shown that in the absence of technological progress a unique steady-state solution with constant prices will exist, it may be asked whether or not a warranted path of capital accumulation from arbitrary initial conditions con- verges to a steady-state path asymptotically. That is to say, is a steady-state path with growth rate ,u stable in the sense of equilibrium dynamics?

Now it has been shown by Hahn that the path traced out by the equilibrium dynamic system will diverge from the steady-state solution for a wide variety of initial conditions.2 In general, therefore, the steady-state solution need not be stable. In particular, if dp/dw is a strictly positive vector, the dual price system cannot be stable for non-zero initial price-expectations, and this instability of the dual leads to instability of the primal. To illustrate tlle instability of the dual, from expression 6 one has

(10) -[dpi (t)/dw (t)] [dw (t)/dri (t)] [dri (t)/dt] - [dp (t)/dt] =pj(t){rX(t) - r(t)}.

If dpi(t)/dw(t) is positive, while of necessity dw(t)/dri(t) is negative, dri(t)/dt must always take the same sign as ri(t) - r(t), all i = 1 ... n, as long as pi(t) is positive. Hence ri(t) will fall if it is less than r(t) and rise if it exceeds r(t). The continuation of such a process cannot lead all rates of return to converge on a common interest rate r(t). It thus cannot lead to a constant price solution. Indeed, such a process may eventually lead to a situation in which some price (s) will become zero, at which point the dynamic equilibrium path ceases to be meaningful and the assumption of perfect foresight can no longer be maintained.3

On the other hand, if the system is allowed to remain on a constant-price ray, then convergence to a steady-state must occur given the capital-intensity condition that dp/dw is strictly positive. For this same capital-intensity condition also implies that dk/dc is strictly positive along any constant-price ray.4 But from expression 6 one has

(11) [dk (t)/dc(t)][dc(t)/dg (t)][dg1(t)/dt] =

[dki(t)/dt] =kj(t){gj(t) - Al.

2Ibid., 642-4. 3See K. Shell and J. E. Stiglitz, "The Allocation of Investment in a Dynamic Economy," Quarterly Journal of Economics, 81 (Nov. 1967), 602-5. 4Compare M. Bruno, "Fundamental Duality Relations in the Pure Theory of Capital and Growth," Review of Economic Studies, 36 (Jan. 1969), 50-2.




Hence if dki(t)/dc(t) is positive, while of necessity along a constant-price ray dc(t)/dg (t) is negative, dg, (t)/dt must always take the opposite sign to g i (t) - u, all i = 1 ... n. Wherever gX(t) exceeds u so that k (t) is rising, gX(t) must fall towards ,u, and wherever g (t) falls short of ,u so that k i(t) is falling, gi(t) niust rise towards ,u, for all i = 1 ... n. The steady-state growth path of the output system on which all quantities grow at the rate ,ut must, in this case, be approached provided that prices can remain constant. Of course, if r (t) must change in order that a steady-state be reached at ,u = sr, prices cannot remain constant.

The basic problem is that if the initial price expectations are given and assumed to be realized then there is nothing to ensure that these expectations, which (because of the perfect foresight assumption) govern the future course of the price system, are appropriately adjusted to the initial stocks in order for convergence to the steady-state solution to occur. Since it is the perfect foresight assumption that all expected capital gains become realized capital gainis that is the cause of instability, this assumption might more appropriately be replaced by an assumption of zero foresight or static expectations whereby no capital gains or losses are expected.5 Prices tomorrow are expected to be the same as prices today, and, if this expectation turns out to have been wrong, decision-making units simply revise their expectations and expect the new prices to continue indefinitely. Expectations are then completely static or of unitary elasticity. If this alternative assumption is made, anticipated capital gains and losses are robbed of any causal significance in the economy. The own-rate of interest equation is abandoned and all rates of return, ri(t), for i = 1 ... n, are taken to be equal to the interest rate, r (t), which is determined directly from g(t) via the savings-investment condition, g(t) = s,r(t). Prices mnove through time in an ex post manner in response to movements in r(t), and, hence, in response to the primal output system. On this assumption, botti the primal and the dual of the Cobb-Douglas dynamic system will normally be stable, converging to the unique steady-state solution dictated by the relation A = srr.6 In particular, if dp/dw is a strictly positive vector the whole Cobb-Douglas dynamic system must be stable on the zero foresight assumption.

IV f Functional forms for technological progress

In this section, technological progress is introduced into the multi-sectoral growth model and alternative specifications of the functional form that technological progress follows are examined. Let it first be assumed that each J =(t), j0, 1 ... n, is a continuous and differentiable function of time t with

the properties

(12) tj(0) = 1, and dtj(t)/dt > 0, for t > 0, all j= 0, 1 ... n. 'Of course, an adaptive expectations hypothesis would be most appropriate in this context; but to incorporate such an hypothesis would complicate the system considerably. 61t is possible to produce some counter-examples in those cases where factor-intensities are perverse. See F. H. Hahn, "Some Adjustment Problems," Econometrica, 38, (Jan. 1970),14-15.



306 B. L. SCARFE

Thus the level of technology in each sector j is a strictly positive and non- decreasing function of time t, while the rate of technological progress in sector j at time t may be defined to be

(13) d ln tj(t)/dt = [1/ij(t)]dtj(t)/dt, j = 0, 1 ... n.

The over-all rate of (Harrod-neutral) technological progress in the economy at time t may be defined by the weighted average

n

(14) Q (t) = v vj(t)d ln tj(t)/dt, j=0

where vj(t) = Lj(t)/L(t) is the proportion of the total labour force that is used in the production of commodity j at time t. It follows that

(15) VI (t) = d ln to(t)/dt + b'w d ln (t)/dt{ (R + D) (G + D)-1 - Bwe}-Ibc(clw),

where w/c = boo + b'wf (R + D) (G + D)-1 - B,-lb,. Alternatively, one may define the over-all rate of technological progress to be the rate of growth of the wage rate w (t) that would occur if R (t) remained constant, which may be obtained from expression 8 as

(16) d ln w(t)/dt = d ln to(t)/dt + b'w d ln (t)/dt{I -Bwc}-'be

Although similar in appearance, these two alternative measures of the over-all rate of technological progress are apparently only identical when there is a common Harrod-neutral rate of technological progress in every sector so that d ln tj(t)/dt = d ln to(t)/dt, all j = 1 ... n, and d ln (t)/dt is the n X n zero matrix. The difference between the two measures is primarily that / (t) uses weights appropriate to the primal quantity equations while the wage rate measure uses weights appropriate to the dual price equations.

The properties given in expression 12 embody an assumption of "societal memory" or "social non-forgetfulness." The stock of technological knowledge that is applied to production processes is assumed never to decline as time passes. It follows immediately that Harrod-neutral technological advances are global in nature, where a global improvement may be defined as one that shifts the factor-price frontier outwards over its whole range allowing a higher real wage rate at each given configuration of net rates of return. In terms of Figure 1, the factor-price frontier shifts upwards in the w (t) direction at every feasible r1 (t), r2 (t) combination.

Although in more general technologies the exact form of the outward shift in the factor-price frontier will normally be fairly complicated, in the Cobb- Douglas case the factor-price frontier shifts outwards in the w(t) direction in a constant proportion at every feasible R (t) configuration. This is, however, simply a consequence of the log-separability of the Cobb-Douglas technology. In more general technologies, a constant proportional shift is not likely to occur except in the special case in which there is a common Harrod-neutral rate of technological progress in every sector at all points of time so that (t) is the n X n identity matrix and one may write the factor-price frontier as w(t) = to(t)Q{R(t)}




It should be evident from the preceding discussion that if technological progress is assumed to occur exponentially at a constant Harrod-neutral rate m1 > 0 in each sector j so that

(17) lj (t) = emit, j = 0, 1 ... n,

then a steady-state solution with constant prices cannot exist except in the very special case where the mj's are identical for all n + 1 sectors. If, then, one is to study the effects of sectorally unbalanced Harrod-neutral technical progress while maintaining the rather useful notion of steady-state equilibrium, the somewhat unrealistic assumption that technical progress occurs exponentially must be abandoned.

Now there are certain continuous functional forms for technological progress which are compatible with at least the asymptotic existence of a steady-state solution. For example, if technological progress in each sector followed the logistic curve common to most diffusion processes, as illustrated (twice) in Figure 2, then an asymptotic steady-state would exist. Mathematically, this may be expressed by

(18) d ln tj(t)/dt = tj(t)/l>(t) = zj{j- (j(t)}l/*j,j = 0, 1 ... n, with solution [t*j - (j(t)]/Ij(t) = {[*j - lj(O)]/Ij(O)}e-zit, j = 0, 1 ...

where zj is a positive constant and where *j > j(0) = 1 is also a positive constant, or, more generally (as illustrated in Figure 2), a non-decreasing step-function of time t. Moreover, there is some realism in this picture of sectoral technological progress as the following argument may suggest.

tj(t)

I Ilnnovational phase two

innovational phase one 1.0

t

FIGURE 2 A logistic functional form

Suppose that it is asked whether a technical improvement is to be associated with (a) an invention, with (b) the first plan to utilize it or the innovation, with (c) the informational diffusion that ensues thereon in which others plan to utilize the invention or variations of it, or with (d) the process of capital transmutation tlhat may be necessary to adapt the economy to the invention. By itself, an invention is not sufficient for a technical change, or a new com-



308 B. L. SCARFE

bination of activities, to be introduced. Moreover, if mistakes can be made and reversed, neither is an innovation, for a reasonable degree of permanence for the new combination of activities may be assumed. On the other hand, if mistakes cannot be made (perfect foresight), innovation will be quickly followed by informational diffusion and the two will always be closely associated. It follows that (a) invention, (b) innovation, and (c) informational diffusion can all be treated as component parts of the notion of a technical improvement.

Informational diffusion is, of course, a time-consuming process. The effect of any particular improvement in a given productive sector is felt gradually as the number of production plants incorporating it expands. The number of production plants introducing the improvement at any particular point of time is, however, likely to follow some fairly regular distribution. Indeed, since it is largely a question of communication, informational diffusion must surely occur in the same sort of way as the spread of any communicable disease through a population. It is this feature which justifies the logistic functional form for each (l(t), though, of course, a further logistic form may follow at a later date as the productive sector responds to a second improvement - a possibility which obviously cannot be ruled out.

Whether or not an improvement must be embodied in new forms of capital goods its implementation will generally require a change in the specification of the economy's capital stock vector, that is some time-consuming process of investment and disinvestment or capital transmutation, which may, of course, occur simultaneously with the process of informational diffusion. The difference between embodied and disembodied progress is simply whether or not the investment part of the process of capital transmutation is connected with the introduction of capital goods whose stock has always been held previously in zero quantities. (From the point of view of analysis this difference is, unfortun- ately, not so simple.) However, in either case, much the same sort of capital transmutation occurs along any non-steady dynamic equilibrium path, whether or not technical progress takes place. It can therefore be argued that in economic terms a technical improvement or a general shift in the state of technical knowledge must be indelibly associated with all three of (a) invention, (b) innovation, and (c) informational diffusion, but that (d) capital transmutation is a distinct but related process whether or not technical progress takes an embodied form.

One of the consequences of the above argument is that it suggests the following analytically convenient simplification.7 Consider an economy in which innovations occur discontinuously as suggested by the t*j step-functions. Suppose, moreover, that the gap between one innovational phase and the next is so long that the economy fully adjusts to one innovational phase before tlle next one is upon it. The economy is in an initial steady-state equilibrium at time t = 0 before the innovation occurs. A single isolated innovation occurs between time t = 0 and time t = 0 + e, but a new steady-state equilibrium is attained before the subsequent innovation occurs. There is, in effect, only

7A similar simplification is made in J. R. Hicks, Capital and Growth (Oxford, 1965), chap. 24, pp. 293-306.




a single "once and for all" step in each step-function *,jj = 0, I ... n, a step which occurs between time t = 0 and time t = 0 + E in all sectors, thiough it does not necessarily affect labour-efficiency in all sectors proportionately.

The assumption of a classical savings function implies that the steady-state rate of interest, r, is uniquely determined by the constant natural rate of growth, ,, via the savings-investment relation, ,u = sr. Hence, if a new steady- state is to be approached after the single discrete innovation occurs, the rate of interest, r(t), must eventually return to its initial steady-state value, r. This implies that the real wage rate must be higher in the new steady-state than in the old. However, in order that a new steady-state will eventually be reached, it must be assumed that the system is stable. Onie convenient way of ensuring that this is so is to assume (a) that expectations are completely static, and (b) that dp/dw is a strictly positive vector. The first of these two assumptions implies that ri(t) = r(t), all i = 1 ... n, so that thie factor-price frontier may be re-defined as w(t)= o(t)Q{r(t), t(t)}, and all motion along the frontier, given to(t) and t(t), is in the single two-dimensional r(t), w(t) plane. The second of these two assumptions implies both uniqueness and stability in the zero foresight or static expectations case, with the consumption good being unambiguously the most capital-intensive conmnodity.

Now suppose for the moment that the process of informational diffusion occurs very quickly relative to the process of capital transmutation. Suppose, indeed, that informational diffusion is virtually instantaneous so that zj -* cc, all j = 0, 1 ... n, and tj(O + e) -* j. Then the two-dimensional factor-price frontier must shift outwards to the right between time t = 0 and time t = 0 + e as illustrated in Figure 3. This shift increases both the rate of inlterest r(t) and the real wage rate w(t) from the initial steady-state point on the old factor-price frontier, say r1, w1 in Figure 3, to a point such as r2, w2 on the new factor-price frontier. The motion from a to d represents this initial shift in factor prices. It therefore represents thle effect of the informatioinal diffusion

w(t)

W3 K

W2 < ^f'

, ~~~new technology

r, 2 r( t)

FIGURE 3 Factor prices and technological change



310 B. L. SCARFE

associated with the technological change. But the economy cannot remain at the point r2, w2 since (given the assumptions made earlier) it must proceed to the new steady-state equilibrium point which is given as the point r1, W3 in Figure 3. Thus the motion from : to y along the new factor-price frontier represents the ensuing time-consuming process of capital transmutation whereby the economy adapts its capital stock position to the technological improvement.

More realistically, however, if informational diffusion is allowed to take time, the shift in the factor-price frontier occurs gradually through time so that the complete motion from a to y follows a path such as the dotted arc in Figure 3, thus smoothing out the kink at A. Nevertheless, it is still possible to keep the effects of informational diffusion conceptually separate from those of capital transmutation in any single innovational phase. The question remains, therefore, what happens to prices and quantities in this combined process of informational diffusion and capital transmutation?

V / Technological progress and capital transmutation

In order to reduce this process to its barest elements, it will be useful to begin with the case in which there are only two commodities, a consumption good (commodity 0) and a capital good (commodity 1). In this case, capital transmutation, if it occurs, must simply take the form of accumulation, altering the capital-labour ratio, k1(t). The capital good is assumed to be the labour- intensive commodity so that its relative price, p'(t), always rises as r(t) falls and w(t) rises along any given technology's factor-price frontier. Since there is no reason to suppose otherwise, it will be convenient to assume that the speed with which informational diffusion occurs is identical in both sectors so that zo = zi = z. There are, then, three basic cases to consider since the technical improvement could have its direct effect on either (a) both sectors in equal proportions, that is %*o = t*i > 1, or (b) the capital-goods producing- sector alone, that is t*% = 1 and t*i > 1, or (c) the consumption-goods producing-sector alone, that is t*o > 1 and t*i = 1.

In all three cases, the following equations must hold at all points at which r (t) and g (t) are constant, namely

(19) d ln w(t)/dt = d ln c(t)/dt = boo d ln to(t)/dt + (1 - boo) d ln 41(t)/dt,

d ln p1(t)/dt = boo d ln to(t)/dt - boo d ln 41(t)/dt, and d ln ki(t)/dt = d ln w(t)/dt-d ln pi(t)/dt = d ln 4i(t)/dt.

It follows immediately from these equations that comparing the new steady- state equilibrium with the old w(t) and c(t) must be higher in all three cases, pi (t) is unchanged in case (a), lower in case (b) and higher in case (c), while k1(t) is higher in cases (a) and (b) but unchanged in case (c). It follows that in cases (a) and (b) capital accumulation must have occurred, so that along the traverse to the new steady-state equilibrium r (t) and g (t) must have been higher than their steady-state values, while p1(t) must have been lower than




its initial value (this last being due, in case (a), to the capital-intensity assumption). On the other hand, in case (c) no capital accumulation occurs, the only effective change being a change in the numeraire. Thus in cases (a) and (b) the path followed is that of the dotted arc from a to y in Figure 3, with r (t) and g (t) rising over the earlier phase of the traverse and falling back to their initial values during its later phase, while in case (c) the path followed is that of the vertical segment a to 7.

Now let the assumption that there is only one capital good be relaxed; in particular, let there be two capital goods (commodity 1 and commodity 2) and a single numeraire consumption good (commodity 0). Given the assumption that the speed with which informational diffusion occurs is the same in all sectors so that ZO = 2"Z2 Z, there are four basic cases to consider depending upon the initial impact of the technological change. These four cases may be listed as case (a) where t*_ = -*l = t*2 > 1, case (b) where *o= t*i = 1 and t*2 > 1, case (c) where t*O= *2 = 1 and t*, > 1, and

case (d) where t*l = t*2 = 1 and t*o > 1. In all four cases, the following equations must hold at all points at which R (t) and G (t) are constant, namely

(20) d In w(t)/dt = d ln c(t)/dt = boo d In to(t)/dt + b'e{I - B'we}-1 d ln ,k (t)/dtbW,

d ln p (t)/dt = [boo d ln to (t)/dt]i - {- ib'e {I - B'wC}-1 d ln ~t(t)/dtbw, and

d ln k (t)/dt = [d ln w (t)/dt]i - d ln p (t)/dt- {I - B we}-' d ln S (t)/dtbw,

where ik(t) -(t)%0(t) is the n X n diagonal matrix with elements (j(t), j = I ... n. It follows immediately from these equations that comparing the new steady-state equilibrium with the old w(t) and c(t) must be higher in all four cases. pi (t) and P2 (t) have settled back to their initial values in case (a), p1(t) is normally higher and P2(t) is lower than initially in case (b), p1(t) is lower and P2(t) is normally higher than initially in case (c), while p1(t) and P2(t) are both higher than initially in case (d), both capital goods having the same proportionate increase in their prices. Notice that it is the capital- intensity hypothesis which explains why p1(t) and P2(t) are normally higher in cases (b) and (c), respectively.

The important point that should be noted, however, comes from the com- parison of capital quantities between the new steady-state and the old. And it is this: unless the technical change generates no substitution effects between the two capital goods, which could occur if either there is no ultimate effect on their relative price as in case (a) and case (d), or there are fixed-coefficients, the new steady-state when compared with the old will have not only different relative prices but also a different combination of capital goods, that is, a different capital stock vector. The proportions in which the two capital goods are held will have changed, as well as their quantities relative to labour (except in case (d) where k1(t) and k2(t) remain unchanged). Capital reallocation will have occurred along the process of adaptation to the technical change. Indeed, although k1(t) and k2(t) will both be higher in the new steady



312 B. L. SCARFE

state than in the old in each of cases (a), (b), and (c), k1(t)/k2(t) will have fallen because of substitution effects if pl(t)/p2(t) has risen, as in case (b), while k1(t)/k2(t) will have risen because of substitution effects if pl(t)/p2(t) has fallen, as in case (c), with k1(t) and k2(t) ultimately rising proportionately in case (a). Capital reallocation is thus a consequence of the substitution effects generated by technical changes.

Along the process of traverse to the new steady-state equilibrium, capital accumulation must occur in all cases except (d) where the only effective change is a change in the numeraire. The initial effects of the technical change are, therefore, to raise the rate of interest, r (t), and the over-all growth rate of the capital stock, g (t), normally by raising both of the individual growth rates, gi(t) and g2(t). On the whole, however, gi(t) must tend to exceed g2(t) along the traverse in case (c) and to fall short of g2(t) along the traverse in case (b). Because of the capital-intensity assumption, p1(t) and P2(t) must both fall in the initial stages of the traverse in case (a), while the way in which pi (t) moves initially in case (b) and the way in which P2(t) moves initially in case (c) are both ambiguous since they depend upon the magnitude of the initial increase in r(t). Eventually, however, g(t) and r(t) must settle back to their initial values so that the traverse again follows a path similar to that of the dotted arc from a to Sy in Figure 3, thereby gradually raising c (t) and w (t) through time.

It should be evident that this manner of analysis can readily be generalized with the aid of equations 20 to the case in which there are many capital goods. It can also be extended to the case in which technological progress takes an embodied instead of a disembodied form as long as one sticks religiously to the putty-putty Cobb-Douglas case in which embodied progress is Harrod- neutral and vintage capital aggregates can be formed.8 Two points should, however, be noted about this extension. First, even if the vintage "productivity improvement factors," 0j(r), i = 1 ... n, are identical for each class of capital goods, the rate of Harrod-neutral technological progress will normally differ from sector to sector since

(21) - ln t(r)b, = B' ln 0(r)i,

where 0(r) is the n X n diagonal matrix of vintage productivity improvement factors, Oi(r), i = 1 ... n, and r represents the vintage number. Secondly, the weighting system based on the Oj(r)'s, which is used in forming the vintage capital aggregates

rt (22) Ki(t)-f KZT(t)0i(r - t) dr, i = 1 ...

_co

will now follow the more complicated logistic pattern rather than the simple exponential pattern used in more traditional models. Fundamentally, however, both the embodied model and the disembodied model illustrate one all-important feature of technical improvements, namely, that through changes in relative prices technical improvements induce substitution effects which necessitate a

80ne version of such an extension can be found in Scarfe, "Capital Accumulation and Com- parative Advantage: A Critical Appraisal," chap. 13, pp. 371-7.




process of capital transmutation and capital reallocation whereby the economy is adapted to the new technology. Capital is continuously being reinvested in more appropriate combinations of capital goods.

VI / Conclusion

This paper has been concerned with the introduction of technological progress into a multi-sectoral model of economic growth. The process of economic growth may be regarded as a process in which the "real" wage rate increases over time, where the "real" wage rate may be measured when there are many consumption goods in terms of some index number of consumption goods prices, such as a Divisia index. However, the process of economic growth is not simply a process of "capital deepening" or accumulation "along the production function" with a declining rate of interest or profit. Fundamentally, capital accumulation occurs in response to increases in the rate of profit that are generated by technological improvements. Since improvements tend to increase the rate of return and accumulation tends to decrease it, a rising rate of return might be observed on average over a period of time in which the effects of technological improvements outstrip the effects of capital accumulation, while a falling rate of return might be observed on average over a period of time in which the effects of technological improvements are outstripped by the effects of capital accumulation. In a monetary economy, the first of these phases might be associated with inflationary periods and the second with deflationary periods or, given sticky money wages and prices, with Keynesian unemployment of both men and machines. In any case, insofar as the effects of technological improvements in increasing the rate of return are not spread randomly through time but are to some degree bunched, the process of growth may in fact follow a cyclical path, growth and cycles thus being intimately related.9

Finally, the "over-all rate of economic growth" in an economy depends very strongly on its ability to introduce technological improvements, and on the responsiveness of capital accumulation to these technological improvements. It is not possible for the "real" wage rate in an economy to rise very rapidly unless both technological improvements and capital accumulation are occurr- ing at a "reasonable pace." For without technological improvements, the rate of return and the pace of accumulation would fall, while without capital accumulation technological improvements of the Harrod-neutral type would simply tend to increase the rate of return. It follows that a very rapid increase in the real wage rate requires both the ability to introduce technological improvements and the ability to invest the resources required to transfer the initial effects of technological irnprovements on the rate of return into resultant effects on the real wage rate.

9Compare J. A. Schumpeter, The Theory of Economic Development (New York, 1961).



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Multi-Sectoral Growth and Technological Change